By carefully choosing the objective functions \(f:2^{{M}}\rightarrow \mathbb {R}\) , we can rely on classical results by G. L. Nemhauser and L. A. Wolsey et al. [1]}, [2]} to prove that greedy algorithms can be used to place the sensors with near-optimal performance.
In particular, each objective that we propose is normalized so that \(f(\emptyset )=0\) , monotone non-decreasing so that \({S}\subseteq {S}^{\prime }\)  implies \(f({S})\le f({S}^{\prime })\) , and has a diminishing returns property called submodularity.
